SIAM J. APPL. MATH.
Vol. 66, No. 2, pp. 533–553
c 2005 Society for Industrial and Applied Mathematics
KINETICS OF MARTENSITIC PHASE TRANSITIONS:
LATTICE MODEL∗
LEV TRUSKINOVSKY† AND ANNA VAINCHTEIN‡
Abstract. Martensitic phase transitions are often modeled by mixed-type hyperbolic-elliptic
systems. Such systems lead to ill-posed initial-value problems unless they are supplemented by an
additional kinetic relation. In this paper we explicitly compute an appropriate closing relation by
replacing the continuum model with its natural discrete prototype. The procedure can be viewed
as either regularization by discretization or a physically motivated account of underlying discrete
microstructure. We model phase boundaries by traveling wave solutions of a fully inertial discrete
model for a bi-stable lattice with harmonic long-range interactions. Although the microscopic model
is Hamiltonian, it generates macroscopic dissipation which can be specified in the form of a relation between the velocity of the discontinuity and the conjugate configurational force. This kinetic
relation respects entropy inequality but is not a consequence of the usual Rankine–Hugoniot jump
conditions. According to the constructed solution, the dissipation at the macrolevel is due to the
induced radiation of lattice waves carrying energy away from the propagating front. We show that
sufficiently strong nonlocality of the lattice model may be responsible for the multivaluedness of
the kinetic relation and can quantitatively affect kinetics in the near-sonic region. Direct numerical
simulations of the transient dynamics suggest stability of at least some of the computed traveling
waves.
Key words. martensitic phase transitions, lattice models, nonlocal interactions, driving force,
lattice waves, radiative damping
AMS subject classifications. 37K60, 74N10, 74N20, 74H05
DOI. 10.1137/040616942
1. Introduction. A characteristic feature of martensitic phase transitions in active materials is the energy dissipation leading to experimentally observed hysteresis.
The dissipation is due to propagating phase boundaries that can be represented at the
continuum level as surfaces of discontinuity. Classical elastodynamics admits nonzero
dissipation on moving discontinuities but provides no information about its origin and
kinetics. Although the arbitrariness of the rate of dissipation does not create problems
in the case of classical shock waves, it is known to be the cause of nonuniqueness in
the presence of subsonic phase boundaries (see [7, 13, 20] for recent reviews).
The ambiguity at the macroscale reflects the failure of the continuum theory to
describe phenomena inside the narrow transition fronts where dissipation actually
takes place. The missing closing relation can be found by analyzing a regularized
theory which describes the fine structure of the transition front. When the local curvature effects can be neglected, the problem reduces to the study of a one-dimensional
steady-state problem. To formulate the simplest problem of this type it is sufficient
to consider longitudinal motions of a homogeneous elastic bar. The total energy of
∗ Received by the editors October 13, 2004; accepted for publication (in revised form) July 22,
2005; published electronically December 30, 2005.
http://www.siam.org/journals/siap/66-2/61694.html
† Laboratoire de Mechanique des Solides, CNRS-UMR 7649, Ecole Polytechnique, 91128,
Palaiseau, France (trusk@lms.polytechnique.fr). The work of this author was supported by NSF
grant DMS-0102841.
‡ Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260 (aav4@pitt.edu).
The work of this author was supported by NSF grant DMS-0137634.
533
534
LEV TRUSKINOVSKY AND ANNA VAINCHTEIN
such bar is the sum of kinetic and potential contributions
2
ρu̇
(1.1)
E=
+ φ(ux ) dx,
2
where u(x, t) is the displacement field, u̇ ≡ ∂u/∂t is the velocity, ux ≡ ∂u/∂x is the
strain, ρ is the constant mass density, and φ(ux ) is the elastic energy density. The
function u(x, t) satisfies the nonlinear wave equation
(1.2)
ρü = (σ(ux ))x ,
where σ(ux ) = φ (ux ) is the stress-strain relation.
Although in classical elastodynamics (1.2), which is often presented as a firstorder p-system, is assumed to be hyperbolic, the hyperbolicity condition σ (ux ) > 0 is
violated for martensitic materials with nonmonotone stress-strain relation σ(ux ) [8].
This makes the initial-value problem associated with the mixed-type equation (1.2)
ill-posed; in particular, it leads to the appearance of discontinuities violating the Lax
condition (subsonic phase boundaries or kinks, e.g., [16, 17, 31]).
To be more specific, consider a discontinuity moving with velocity V . Let f−
and f+ denote the limiting values of a function f (x) to the left and to the right of
the interface, and introduce the notations [[f ]] ≡ f+ − f− for the jump and {f } ≡
(f+ + f− )/2 for the average of f across the discontinuity. The parameters on a
discontinuity must satisfy both the classical Rankine–Hugoniot jump conditions,
(1.3)
[[u̇]] + V [[ux ]] = 0,
ρV [[u̇]] + [[σ(ux )]] = 0,
and the entropy inequality R = GV ≥ 0, where
(1.4)
G = [[φ]] − {σ}[[ux ]]
is the configurational (driving) force. Contrary to conventional shock waves, the
martensitic phase boundaries usually fail to satisfy the Lax condition c+ < V < c− ,
where c± are the values of the sound velocity in front and behind the discontinuity.
One way to remedy the resulting nonuniqueness is to supplement (1.3) by a kinetic relation specifying the dependence of the configurational force on the velocity
of the phase boundary G = G(V ) [1, 30]. Since the nonlinear wave equation (1.2)
provides no information about the kinetic relation, the dependence G(V ) has often
been modeled phenomenologically [1, 29, 30]. An alternative approach has been to
derive the kinetic relation from an augmented model incorporating regularizing terms.
A typical example is the viscosity-capillarity model, accounting for both dispersive
and dissipative corrections [21, 29]. The problem with both approaches is that they
introduce into the theory parameters of unclear physical origin.
The aim of the present paper is to obtain the kinetic relation without any phenomenological assumptions at the macroscale by means of direct replacement of the
continuum model (1.2) with its natural discrete prototype. Such procedure of going
back from continuum to discrete level can be viewed as either regularization by discretization or as a physically motivated account of underlying atomic or mesoscopic
microstructure. It is clear that the discrete model must be Hamiltonian to reproduce
the conservative structure of the smooth solutions of (1.2). The energy dissipation on
the discontinuities can then be interpreted as the nonlinearity-induced radiation of
lattice-scale waves which takes the energy away from the long-wave continuum level.
This phenomenon is known in physics literature as radiative damping (e.g., [11, 12]).
KINETICS OF PHASE TRANSITIONS
535
To regularize (1.2) from “first principles,” we consider in this paper fully inertial
dynamics of a one-dimensional lattice with bi-stability and long-range interactions.
Following some previous work on cracks [22] and dislocations [2], we assume piecewise
linear approximation of nonlinearity and construct an explicit traveling wave solution
of the discrete problem. There exists an extensive literature on shock waves and
solitons in the local and nonlocal discrete systems with convex interatomic potentials
(e.g., [9, 10, 19, 27]) and on the semilinear prototypes of the present bi-stable system
(e.g., [3, 5, 18]). A discrete quasilinear problem for martensitic phase transitions and
failure waves in the chains with nearest-neighbor (NN) interactions was considered in
[24, 25, 26, 35]. In the present paper we extend these results to the case of harmonic
interactions of finite but arbitrary long range. We show that the local (NN) model
is degenerate, find a general solution of the nonlocal model, and provide detailed
illustrations for a particular case.
Our analytic solutions demonstrate that the nonlocal model generates a much
broader class of admissible solutions than the local model; in particular, it allows the
possibility of radiation both in front and behind the moving discontinuity. We also
show that sufficiently strong nonlocality may be responsible for the multivaluedness of
the kinetic relation and can quantitatively affect kinetics in the near-sonic region. The
advantage of the explicit formulas obtained in the paper is that they capture certain
details that are difficult or even impossible to detect in numerical simulations, such as
singular behavior of solutions near static-dynamic bifurcation and around resonances.
The paper is organized as follows. The piecewise linear discrete model with longrange interactions and the associated dynamical system are introduced in section 2.
In section 3 we formulate the dimensionless equations for the traveling waves, the
boundary conditions, and the admissibility conditions. An explicit solution for the
steady state motion of an isolated phase boundary is obtained by Fourier transform
in section 4. In section 5 we obtain static solutions describing lattice-trapped phase
boundaries and link them to a nontrivial limit of the dynamic solutions. The energy
transfer from long to short waves is studied in section 6, where we obtain a closedform kinetic relation. In section 7 we illustrate the general theory via the case when
the only long-range interactions are due to the second nearest neighbors. Numerical
simulations of the transient problem suggesting stability of at least sufficiently fast
traveling waves are described in section 8. The last section contains our conclusions.
2. Discrete model. The simplest lattice structure can be modeled as a chain
of point masses connected by elastic springs. Suppose that the interactions are of
long-range type and that every particle interacts with its q neighbors on each side.
If un (t) is the displacement of the nth particle, the total energy of the chain can be
written as
(2.1)
E =ε
q
∞ ρu̇2n un+p − un
+
,
pφp
2
pε
n=−∞
p=1
where ε is the reference interparticle distance and φp (w) is the energy density of the
interaction between pth nearest neighbors. The dynamics of the chain with energy
(2.1) is governed by an infinite system of ordinary differential equations:
(2.2)
q un − un−p
1 un+p − un
− φp
.
φ
ρün =
ε p=1 p
pε
pε
536
LEV TRUSKINOVSKY AND ANNA VAINCHTEIN
A continuum system, formally obtained by identifying u(x, t) with a limit of u(nε, t) =
un (t) as ε → 0, reduces to the nonlinear wave equation (1.2) with a specific macroscopic stress-strain relation
σ(w) =
(2.3)
q
pφp (w).
p=1
When the function σ(w) is nonmonotone, (1.2) constitutes an incomplete description
of the limit. As we show, in this case the correct limit procedure starting from the
discrete problem (2.1), (2.2) must also produce a specific kinetic relation G = G(V ).
This relation, expressed exclusively in terms of the elastic potentials entering (2.1),
provides the desired closure for the macroscopic problem (1.2), (1.3). Here we do not
consider the issue of a nucleation criterion, whose discrete prototype was studied in
[17].
To obtain analytical results, we consider the simplest potentials allowing for a possibility of a phase transitions: bi-quadratic for local interactions (NN) and quadratic
for nonlocal interactions (NNN, NNNN, etc.). Specifically we define
(2.4)
φ1 (w) =
⎧ 1
2
⎪
⎪
⎨ 2 Ψ(1)w ,
w ≤ wc ,
⎪
⎪
⎩ 1 Ψ(1)(w − a)2 + aΨ(1) wc − a , w ≥ wc ,
2
2
and
φp (w) =
(2.5)
1
pΨ(p)w2 , p = 2, . . . , q.
2
One can see that the nonlinear springs representing NN interactions can be found in
two different states depending on whether the strain w is below (phase I) or above
(phase II) the critical value wc . Parameter a defines the microscopic transformation
strain (distance between the two linear branches); note that a and wc are in general
independent. For simplicity we assume that the two energy wells of the bi-stable NN
potential have equal curvatures Ψ(1) > 0.
It is convenient to reformulate the problem using dimensionless variables:
(2.6)
t̄ = t(Ψ(1)/ρ)1/2 /ε,
u¯n = un /(aε),
w̄c = wc /a,
Ψ̄(p) = Ψ(p)/Ψ(1), p = 1, . . . , q.
In terms of these variables with the bars dropped, the energy (2.1) becomes
(2.7)
E=
∞ 2
u̇n
1 un Ψ(k − n)uk − (un − un−1 − wc )θ(un − un−1 − wc ) .
−
2
2
n=−∞
|k−n|≤q
By introducing the strain variables wn = un − un−1 , we can rewrite the governing
equations (2.2) in the form
(2.8) ẅn −
Ψ(k − n)wk = 2θ(wn − wc ) − θ(wn+1 − wc ) − θ(wn−1 − wc ),
|k−n|≤q
537
KINETICS OF PHASE TRANSITIONS
I
phase II
A1
A2
phase I
w+
wc
w−
w
Fig. 2.1. The bi-linear macroscopic stress-strain law and the Rayleigh line connecting the
states at infinity for a traveling wave solution describing an isolated phase boundary. The difference
between the shaded areas A2 − A1 represents the configurational force.
where
(2.9)
Ψ(0) = −2
q
Ψ(p), Ψ(−p) = Ψ(p),
p=1
and θ(w) is a unit step function. The macroscopic stress-strain relation (2.3) takes
the form
σ(w) = c2 w − θ(w − wc ),
(2.10)
where
c=
(2.11)
q
1/2
p2 Ψ(p)
p=1
is the dimensionless macroscopic sound speed. The microscopic elastic moduli Ψ(p)
must be chosen to ensure that the uniform deformation wn = w is stable in each of the
phases. For this it is necessary and sufficient that all phonon frequencies ω 2 (k) > 0,
with ω(k) defined in (3.6) and k ∈ (0, π], are real. This implies, in particular, that the
square of the macroscopic sound speed (2.11) is positive. The resulting macroscopic
stress-strain relation (2.10) is shown in Figure 2.1.
3. Traveling waves. An isolated phase boundary moving with a constant velocity V can be obtained as a traveling wave solution of (2.8) with wn (t) = w(ξ),
ξ = n − V t. We assume further that in the moving coordinate system all springs in
the region ξ > 0 are in phase I (wn < wc ), and all springs with ξ < 0 are in phase II.
The system (2.8) can then be replaced by a single nonlinear advance-delay differential
equation:
V 2 w −
(3.1)
Ψ(p)w(ξ + p) = 2θ(−ξ) − θ(−ξ − 1) − θ(1 − ξ).
|p|≤q
The configurations at ξ = ±∞ must correspond to stable homogeneous equilibria
plus superimposed short-wave oscillations with zero average; the averaging is over the
largest period of oscillations but can be also defined as
1 ξ+s
w(ξ) = lim
(3.2)
w(ζ) dζ.
s→∞ s ξ
538
LEV TRUSKINOVSKY AND ANNA VAINCHTEIN
In terms of the averaged quantities we obtain the following boundary conditions:
w(ξ) → w±
(3.3)
as ξ → ±∞.
The nonlinearity of the problem is in the switching condition
w(0) = wc .
(3.4)
We assume that a solution is admissible if the NN springs in front of the moving
interface are still in phase I and behind it already in phase II. This implies that
(3.5)
w(ξ) < wc
for ξ > 0,
w(ξ) > wc
for ξ < 0.
Consequently, the mathematical problem reduces to solving (3.1) subject to (3.3),
(3.4), and (3.5).
Observe first that the equation (2.8) is linear in each phase (ξ < 0 and ξ > 0),
which means that the solution can be represented as a superposition of linear waves
wn = exp(i(kn − ωt)). Since the elastic moduli are equal, the dispersion relation
(3.6)
ω 2 (k) = 4
q
Ψ(p) sin2
p=1
pk
2
is the same in both phases. For the linear modes to be compatible with the traveling
wave ansatz, their phase velocity Vp (k) = ω/k must be equal to V . This gives the
restriction on the admissible wave numbers in the form
(3.7)
L(k, V ) = 0,
where
(3.8)
L(k, V ) = 4
q
p=1
Ψ(p) sin2
pk
− V 2 k2 .
2
Among the modes selected by (3.7), the ones with complex wave numbers must be
exponentially decaying on both sides of the front. They describe the core structure
of the phase boundary. The modes with nonzero real wave numbers correspond to
radiation. The waves with k = 0 are naturally associated with the macroscopic part
of the solution.
4. Exact solution. We solve (3.1) by writing w(ξ) = h(ξ) + w− and applying
the complex Fourier transform
∞+iα
∞
1
i(k+iα)ξ
h(ξ)e
dξ, h(ξ) =
ĥ(k)e−ikξ dk,
ĥ(k) =
2π −∞+iα
−∞
where α > 0 is a small parameter which guarantees convergence of the integrals. After
inverting the Fourier transform and letting α → 0, we obtain
sin2 (k/2)eikξ dk
2
w(ξ) = w− −
(4.1)
,
πi Γ
kL(k, V )
where the contour Γ coincides with the real axis passing the singular point k = 0
from below. The singularities associated with nonzero real roots of L(k, V ) = 0 must
KINETICS OF PHASE TRANSITIONS
539
comply with the radiation conditions. Specifically, the modes with group velocity
Vg = ∂ω/∂k larger than V can appear only in front, while the modes with Vg < V
can appear only behind the phase boundary [24]. Using the relation
(4.2)
Vg = V +
Lk (k, V )
,
2V k
where Lk (k, V ) = ∂L/∂k and assuming V > 0, we obtain that Vg ≷ V whenever
kLk (k, V ) ≷ 0. Therefore, to satisfy the radiation conditions, we need to dent the
integration contour in (4.1) in such a way that it passes below the singularities on the
real axis if kLk (k, V ) > 0 and above if kLk (k, V ) < 0.
To compute the integral (4.1) explicitly, we use the residue method closing the
contour in the upper half-plane when ξ > 0 and in the lower half-plane when ξ < 0.
The solutions look different in the generic case q > 1 and the degenerate case q = 1.
For q > 1 the Jordan lemma can be applied directly, and by separating the
macroscopic part of the solution from the microscopic one, we obtain
⎧
4 sin2 (k/2)eikξ
⎪
⎪ w− +
for ξ < 0,
⎪
⎨
kLk (k, V )
k∈M − (V )
(4.3)
w(ξ) =
⎪
4 sin2 (k/2)eikξ
1
⎪
⎪
for ξ > 0.
−
⎩ w− − 2
c − V 2 k∈M + (V ) kLk (k, V )
Here
(4.4)
M ± (V ) = {k : L(k, V ) = 0, Imk ≷ 0}
N ± (V )
are all roots of the dispersion relation contributing to the solution on either side of
the front, with
(4.5)
N ± (V ) = {k : L(k, V ) = 0, Imk = 0, kLk (k, V ) ≷ 0}
denoting the sets of real roots describing radiation.
For q = 1 (NN interactions only) the contribution from a semi-arch at infinity
does not vanish at ξ = ±0 and relations (4.3) must be supplemented by the following
limiting conditions:
⎧
4 sin2 (k/2)eikξ
1
⎪
⎪
w
−
for ξ = −0,
+
−
⎪
⎨
kL
(k,
V
)
2
−
k
k∈M (V )
(4.6)
w(ξ) =
⎪
4 sin2 (k/2)eikξ
1
1
⎪
⎪ w− −
+
for ξ = +0.
−
⎩
c2 − V 2 k∈M + (V ) kLk (k, V )
2
In both cases, by applying the boundary conditions (3.3) at infinity we obtain
(4.7)
w+ = w− −
1
.
c2 − V 2
It is easy to see that (4.7) coincides with the Rankine–Hugoniot relation V 2 [[w]] = [[σ]],
computed for the macroscopic stress-strain relation (2.10). The continuity of w(ξ) at
ξ = 0 implies that
4 sin2 (k/2) 1, q = 1,
1
(4.8)
=
+
0, q > 1,
c2 − V 2
kLk (k, V )
k∈M (V )
540
LEV TRUSKINOVSKY AND ANNA VAINCHTEIN
where M (V ) = M + (V ) M − (V ). Condition (4.8) is automatically satisfied for q > 1
since the sum of residues at all poles (including k = 0) equals zero; for q = 1 and ξ = 0
the integral over a contour at infinity contributes additional unity in the right-hand
side of (4.8). The switching condition (3.4) together with (4.8) requires that
(4.9)
w± = wc ∓
1
+
2(c2 − V 2 )
k∈Npos (V )
4 sin2 (k/2)
,
|kLk (k, V )|
where Npos (V ) = {k : L(k, V ) = 0, Imk = 0, k > 0} ⊂ M (V ) is the set of positive
real roots of the dispersion relation. By virtue of (4.7), the two conditions (4.9) are
dependent and can be replaced by a single condition:
(4.10)
1
(w− + w+ ) − wc =
2
k∈Npos (V )
4 sin2 (k/2)
.
|kLk (k, V )|
As we show in section 6, (4.10) represents the desired kinetic relation.
We can use the explicit formulas for w(ξ) to reconstruct the particle velocity
profile from v(ξ) = −V u (ξ) if we assume that V = 0. The relation between the
velocity and the strain fields reads
(4.11)
v(ξ) − v(ξ − 1) = −V w (ξ),
and the right-hand side is already known from (4.3). Solving (4.11) by Fourier transform, we obtain
⎧
1
sin(k/2)eik(ξ+ 2 )
⎪
1
V
⎪
⎪ v+ −
for ξ < − ,
−
2V
⎪
2
2
⎨
c −V
2
Lk (k, V )
k∈M − (V )
(4.12)
v(ξ) =
1
ik(ξ+
)
⎪
2
sin(k/2)e
1
⎪
⎪
for ξ > − .
⎪
⎩ v+ + 2V
L
(k,
V
)
2
k
k∈M + (V )
It is easy to check that the average velocities at infinity satisfy the remaining Rankine–
Hugoniot condition (1.3)1 , which in our case takes the form
v+ − v− =
(4.13)
c2
V
.
−V2
Notice that the obtained set of traveling wave solutions is parametrized by the
velocity V and the boundary value data w± and v± . The average particle velocity
v+ in front can always be set equal to zero due to the Gallilean invariance. If the
strain in front of the discontinuity is also prescribed, the remaining three macroscopic
parameters are fully constrained by the two classical Rankine–Hugoniot conditions
(4.7) and (4.13), plus the nonclassical admissibility condition (4.10).
5. Static solutions. A special consideration is needed when V = 0. In this
case continuous variable ξ = n − V t takes integer values, and the strain profile becomes discontinuous at every ξ = n. The differential equation reduces to a system of
finite-difference equations, and we can replace the continuous Fourier transform by its
discrete analogue (see also [6, 11, 23]). First observe that for a piecewise continuous
function w(ξ) with discontinuities at integer ξ we have
∞
∞ n+1
ŵ(k) =
w(ξ)eikξ dξ =
w(ξ)eikξ dξ.
−∞
n=−∞
n
KINETICS OF PHASE TRANSITIONS
541
Therefore, assuming that the strain profile w(ξ) converges to w0 (n) as V → 0, we
obtain
(5.1)
ŵ0 (k) =
∞
w0 (n)
n=−∞
eik − 1 D
eik(n+1) − eikn
=
ŵ0 (k),
ik
ik
∞
where ŵ0D (k) = n=−∞ w0 (n)eikn is the discrete Fourier transform of w0 (n). Now
we can use (4.1) to obtain
ŵ0 (k) = 2πδ(k)w− +
4 sin2 (k/2)
,
ikω 2 (k)
where δ(k) is the Dirac delta function and ω 2 (k) is given by (3.6). Using (5.1) we can
then find ŵ0D (k) and, applying inverse discrete Fourier transform, obtain a representation of the discrete solution:
π
π
sin(k/2)eik(n+1/2) dk
1
1
D
−ikn
.
ŵ0 (k)e
wn =
dk = w− −
2π −π
πi −π
ω 2 (k)
To avoid the singularity at k = 0 we must pass it from below; all other roots of
the equation ω 2 (k) = 0 inside the strip −π ≤ Rek ≤ π have nonzero imaginary
parts. Closing the contour of integration in the upper half-plane for n ≥ 0 and lower
half-plane for n < 0, we obtain by residue theorem
⎧
sin(k/2)eik(n+1/2)
⎪
⎪
,
n < 0,
w
+
⎪
−
⎨
ω(k)ω (k)
k∈F −
wn =
(5.2)
ik(n+1/2)
⎪
⎪
⎪ w− − 1 − sin(k/2)e
, n ≥ 0,
⎩
2
c
ω(k)ω (k)
k∈F +
where F ± = {k : ω 2 (k) = 0, Imk ≷ 0, −π ≤ Rek ≤ π}. Solutions satisfying the
admissibility constraints
(5.3)
wn ≥ wc
for n ≤ −1,
wn ≤ wc
for n ≥ 0
form a family of lattice-trapped equilibria parametrized by the total stress in the chain
σ = c2 w− − 1; the set of such stresses constitutes the trapping region.
To specify the trapping region we observe that in our static solutions the phase
boundary is pinned at the site n = −1. If the strain profile (5.2) is monotone, which
occurs, for example, when all long-range interactions are repulsive (Ψ(p) < 0 for
p ≥ 21 ), the constraints (5.3) can be replaced by w0 ≤ wc and w−1 ≥ wc . The
trapping region can then be described explicitly:
(5.4)
σM − σP ≤ σ ≤ σM + σP ,
where σM = c2 wc − 1/2 is the Maxwell stress and
(5.5)
σP =
sin(k/2)eik/2
1
+ c2
2
ω(k)ω (k)
+
k∈F
1 Since Ψ(1) = 1 > 0, the homogeneous phases can still be stable if the negative long-range
moduli are sufficiently small.
542
LEV TRUSKINOVSKY AND ANNA VAINCHTEIN
is the Peierls stress (see also [4, 32]). The phase boundary remains trapped until the
stress reaches one of the limiting values: σ = σM − σP , corresponding to w−1 = wc
when the interface starts moving to the left (V < 0), or σ = σM + σP , corresponding
to w0 = wc when the interface starts moving to the right (V > 0). The two limiting
solutions represent unstable equilibria from which the dynamic solution bifurcates.
6. Kinetic relation. The waves generated in the core of the moving phase
boundary carry the energy away from the front without changing the average values
of parameters at infinity. At the continuum level these lattice waves are invisible and
therefore the associated energy transfer is perceived as dissipation. To evaluate the
rate of dissipation, we start with the microscopic energy balance
dE
= A(t),
dt
where E is the total energy of the chain and A(t) is the power supplied by the external
loads. Since the solution of the discrete problem at infinity can be represented as a
sum of the macroscopic contribution and the superimposed oscillations, we can split
the averaged power accordingly. We obtain
A = P − R,
(6.1)
where P = σ+ v+ − σ− v− is the macroscopic rate of work and R is the energy release
due to radiated waves which is invisible at the macroscale. While in the general
case the expression for R may contain coupling terms, in the piecewise linear model
adopted in this paper, the macroscopic and microscopic contributions decouple (see
also [11, 24]). The dissipation rate R can be written as the sum of the contributions
from the areas ahead and behind the front:
R(V ) = R+ (V ) + R− (V ).
(6.2)
To specify the entries in the right-hand side, we observe that due to the exponential
decay of the modes with complex wave numbers, the strain and velocity fields given
by (4.3) and (4.12) have the following asymptotic representation at ξ = ±∞:
v(ξ) ≈ v0 (ξ) +
vk (ξ), w(ξ) ≈ w0 (ξ) +
wk (ξ),
k∈Npos (V )
where
v0 (ξ) =
v− ,
v+ ,
ξ < 0,
ξ > 0,
k∈Npos (V )
w0 (ξ) =
w− ,
w+
ξ < 0,
ξ > 0,
are the homogeneous components and
⎧
4V sin(k/2) cos(k(ξ − 1/2))
⎪
−
⎪
(V ),
, ξ < 0, k ∈ Npos
⎪
⎨ −
Lk (k, V )
vk (ξ) =
⎪
⎪
+
⎪ 4V sin(k/2) cos(k(ξ − 1/2)) ,
ξ > 0, k ∈ Npos
(V ),
⎩
Lk (k, V )
(6.3)
⎧
8 sin2 (k/2) cos kξ
⎪
−
⎪
(V ),
,
ξ < 0, k ∈ Npos
⎪
⎨
kLk (k, V )
wk (ξ) =
⎪
8 sin2 (k/2) cos kξ
⎪
+
⎪
⎩ −
(V ),
, ξ > 0, k ∈ Npos
kLk (k, V )
543
KINETICS OF PHASE TRANSITIONS
±
(V ) ≡ {k ∈ N ± (V ) : k > 0} (recall (4.5)),
are the oscillatory
components. Here Npos
−
+
Npos (V ) Npos (V ) = Npos (V ). Due to the asymptotic orthogonality of the linear
modes, the terms in the right-hand side of (6.2) can be expressed as contributions due
to individual modes. Since the energy flux associated with the linear mode k is the
product of the average energy density Gk and the relative velocity |Vg − V | of the
energy transport with respect to the moving front, we can write
R+ (V ) =
(6.4)
Gk + (Vg − V ),
R− (V ) =
Gk − (V − Vg ).
−
k∈Npos
(V )
+
k∈Npos
(V )
Here Gk ± is the average energy density carried by the wave with the wave number
±
k ∈ Npos
(V ). It is given by
n
1
vk2 (ξ) + c2 (wk (ξ))2
Gk ± = lim
n→±∞ 2V τ (k) n−V τ (k)
−
q−1
B(p){(wk (ξ + p) − wk (ξ)) + (wk (ξ) − wk (ξ − p)) } dξ,
2
2
p=1
where B(p) =
obtain
q−p
1
2
l=1
lΨ(l + p) and τ (k) = 2π/ω(k) = 2π/(V k). Using (6.3), we
Gk ± =
8V 2 sin2 (k/2)
,
(Lk (k, V ))2
which gives for the total energy flux
R(V ) =
+
k∈Npos
(V )
4V sin2 (k/2)
−
kLk (k, V )
−
k∈Npos
(V )
4V sin2 (k/2)
=
kLk (k, V )
k∈Npos (V )
4V sin2 (k/2)
.
|kLk (k, V )|
Recalling the definition R(V ) = G(V )V we can write the microscopic expression for
the configurational force:
(6.5)
G(V ) =
k∈Npos (V )
4 sin2 (k/2)
.
|kLk (k, V )|
The function G(V ) is well-defined since both L(k, V ) and Npos (V ) depend on V in a
known way. Comparing (6.5) with the macroscopic definition of the configurational
force (1.4) we obtain
(6.6)
G=
1
(w− + w+ ) − wc ,
2
which can be interpreted geometrically as the area difference between two shaded
triangles in Figure 2.1. Combining (6.5) and (6.6), we obtain exactly (4.10), which
shows that (4.10) is indeed the desired kinetic relation and that micro and macro
assessments of dissipation are compatible.
7. An example. To illustrate the general solution, consider a special case q = 2
(see Figure 7.1). The model is then fully characterized by a single dimensionless
parameter,
β = 4Ψ(2)/Ψ(1),
544
LEV TRUSKINOVSKY AND ANNA VAINCHTEIN
bi-stable spring
−3
−2
linear spring
−1
0
1
2
3
ε
2ε
Fig. 7.1. Discrete chain with nearest and next-to-nearest-neighbor interactions (q = 2).
measuring the relative strength of NNN and NN interactions. The stability constraints
(e.g., [36]) give −1 < β ≤ ∞. Recalling that the inequality Ψ(2) < 0 is suggested by
the linearization of the potentials of the Lennard–Jones type [32, 33], we can further
restrict the admissible interval to
(7.1)
−1 < β ≤ 0.
The total energy of the system can be written as
∞ 2
1+β 2
vn
β
2
E=
+
wn − θ(wn − wc )(wn − wc ) − (wn+1 − wn ) .
(7.2)
2
2
4
n=−∞
One can see that parameter β/(1 + β) characterizes the effect of discreteness: β ∼ 0
corresponds to weak and β ∼ −1 to strong coupling. This identification is compatible
with the fact that at β = 0 the Peierls stress characterizing the size of the latticetrapping domain takes the largest value (equal to the spinodal limit), while at β = −1
the Peierls stress is equal to zero.2
The energy (7.2) produces the following equation for the traveling waves:
β
w(ξ + 2) − 2w(ξ) + w(ξ − 2) − w(ξ + 1) + 2w(ξ) − w(ξ − 1)
V 2 w −
4
(7.3)
= 2θ(−ξ) − θ(−ξ − 1) − θ(1 − ξ).
The formal solution of this equation has been obtained in section 4. Below we provide
detailed illustrations for the physically relevant range of parameters β.
7.1. Dispersion relation. To compute the strain and velocity profiles at a given
V we need to find the nonzero roots k of the dispersion relation
(7.4)
L(k, V ) = 4 sin2 (k/2) + β sin2 k − V 2 k 2 = 0.
It is convenient to present the complex roots explicitly as k = k1 +ik2 and divide them
into three categories: real, responsible for radiation; purely imaginary, providing the
monotone structure of the core region; and complex with nonzero real part, describing
oscillatory contributions to the core.
Since L(k, V ) is an even function of k, the real roots appear in pairs k = ±k1 .
Assuming positive V , we obtain
4 sin2 (k1 /2) + β sin2 k1
.
V (k1 ) =
|k1 |
This function is plotted in Figure 7.2(a). An infinite number of local maxima on
2 The picture emerging in our piecewise linear model is somewhat obscured by the fact that in
the limit of strong coupling the macroscopic sound speed tends to zero.
545
KINETICS OF PHASE TRANSITIONS
(b)
(a)
b=0
V
0.8
b=0
V
b = - 0.25
1.2
b = - 0.6
1
0.6
0.8
0.4
0.6
0.2
b = - 0.25
0.4
b=-1
0.2
5
10
15
k1
b = - 0.6
b = - 0.95
0.5
1
1.5
2
2.5
k2
Fig. 7.2. Real (a) and imaginary (b) roots of the dispersion relation L(k, V ) = 0 at different β.
this graph, denoted by V = Vi , correspond to resonance velocities: at these points
Lk (k, V ) = 0 and the sums in (4.3), (4.9), and (4.12) diverge. Between the resonance
velocities, (7.4) possesses a finite number of positive real roots corresponding to propagating waves. To determine whether these waves propagate ahead or behind the
front, we need to check whether kLk (k, V ) = 2k 3 V (k)V (k) is positive or negative.
At V > 0 the radiation conditions say that the waves with kV (k) > 0 propagate in
front of the phase boundary, while the waves with kV (k) < 0 propagate behind.
Changing β affects the function V (k) noticeably only at long waves (small k).
A straightforward computation shows that V (0) is equal to the macroscopic sound
speed c = (1 + β)1/2 . We also obtain that V (0) = 0 and
V (0) = −
1 + 4β
√
.
12 1 + β
At −1/4 < β ≤ 0, the function V (k) has a maximum at k = 0 while at −1 < β < −1/4
it has a local minimum implying that sufficiently strong coupling (β < −1/4) creates
the possibility for the lattice waves to move faster than the macroscopic sound speed.
The range of supersonic speeds increases as β → −1, and in the limiting case β = −1
all propagating waves are macroscopically supersonic. It is interesting that the critical
value β = −1/4 also emerges in the strain-gradient approximation of the energy
(7.2), where it corresponds to the change of sign of the coefficient in front of the
strain gradient term [15, 28]. In this approximation the dispersion relation V (k) is
replaced by a parabola: for β > −1/4 (weak nonlocality) the parabola is directed
downward and the strain-gradient coefficient is negative while for β < −1/4 (strong
nonlocality) the parabola is upward and the strain-gradient contribution to the energy
is positive definite. The latter implies that subsonic phase boundaries can only be
dissipation free. To yield a nontrivial kinetic relation in this range of parameters the
quasicontinuum model must be augmented by higher-order terms [37].
The purely imaginary roots of (7.4) appear in symmetric pairs and correspond
to nonoscillatory modes exponentially decreasing away from the front. By solving
L(ik2 , V ) = 0 for V we obtain
4 sinh2 (k2 /2) + β sinh2 k2
V (k2 ) =
.
|k2 |
This function is shown in √Figure 7.2b. One can show that V (0) = c, V (0) = 0,
and V (0) = (1 + 4β)/(12 1 + β). For −1 < β < −1/4 the maximum of the curve
546
LEV TRUSKINOVSKY AND ANNA VAINCHTEIN
b = -1/16
b = -2/3
6
5
k2
4
k2
3
2
4
2
1
0
1.5
0
1
0.8
1
V
V 0.6
0.4
0.5
0.2
0
0
0
0
5
10
10
k1
k1
15
20
20
25
Fig. 7.3. The structure of nonzero roots of L(k, V ) = 0 in the cases of strong (β = −2/3)
and weak (β = −1/16) nonlocality. Thin lines: real roots, set N ; thick lines: P -roots; gray lines:
Q-roots.
V (k2 ) is reached at k2 = 0, which means that in the case of strong coupling only
macroscopically subsonic phase boundaries have monotone contribution to the core
structure. Both the value V (0) and the range of available wave numbers decrease as
β → −1, so that in the limit purely imaginary roots disappear. In the case β = 0 the
function V (k2 ) is convex and no imaginary roots contribute to the subsonic solution.
This is compatible with the fact that in the degenerate NN limit the static interface
(V = 0) is atomically sharp.
Complex roots with nonzero real part contribute to the oscillatory structure of
the core region. For the given V the real (k1 ) and imaginary (k2 ) parts of the relevant
wave numbers satisfy the system of two equations: ReL(k1 +ik2 , V ) = 0 and ImL(k1 +
ik2 , V ) = 0. The set of complex roots contains infinitely many branches that come in
symmetric quadruples. The first quadrant of the complex plane is shown in Figure 7.3.
The complex roots can be divided into two sets: Q and P . The set Q (thick gray
lines), has a purely dynamic nature and contributes to the boundary layers around
the front only at nonzero V . The set P , shown in Figure 7.3 by thick black lines,
contains purely imaginary branches which intersect the plane V = 0 and contribute
to the static solution: at V = 0 they are given by
1
k = 2πn ± iλ,
λ = 2arccosh (7.5)
,
|β|
where n is an integer [33]. As β tends to zero, the imaginary parts of P -roots approach
±∞; the eventual disappearance of these roots in the limit β → 0 is responsible for
the sharpening of the front in the NN approximation.
7.2. Strain and velocity profiles. Typical profiles of strain w(ξ) and velocity v(ξ) computed for the NNN model from (4.3), (4.12) are shown in Figure 7.4,
where β = −0.2. For this case the first two resonance velocities are V1 = 0.2164
and V2 = 0.1282. Accordingly, at V = 0.5 > V1 we see only one radiative mode
propagating behind the phase boundary; at V2 < V = 0.16 < V1 the solution exhibits
two additional radiative modes, one propagating behind and one in front of the phase
boundary.
547
KINETICS OF PHASE TRANSITIONS
V = 0.16
V = 0.5
w
w
2.5
2.5
2
2
1.5
1.5
phase II
1
1
phase I
0.5
0.5
-5
0
5
N
v
-5
0
5
N
-5
0
5
N
v
0
0
-0.5
-0.5
-1
-1
-1.5
-1.5
-5
0
5
N
Fig. 7.4. Strain and velocity profiles at V > V1 and V2 < V < V1 . Here β = −0.2, wc = 1,
v+ = 0.
V = 0.16
V = 0.105
w
w
3
3
2
2
phase II
1
0
1
phase I
0
-1
-1
-5
0
5
N
-5
0
5
N
Fig. 7.5. Strain profiles at V2 < V < V1 and V3 < V < V2 . Here β = −0.75, wc = 1.
A closer inspection of the solutions at V < 0.266 reveals a violation of the constraints (3.5): in the strain profile corresponding to V = 0.16 in Figure 7.4 the
threshold w = wc is crossed at both ξ = 0 and ξ > 0. Moreover, for this value of β
our numerical computations suggest that the entire velocity interval (0, 0.266) around
the resonances has to be excluded as inadmissible. Similar “velocity gaps” also have
been detected in [11, 12, 14] for the semilinear Frenkel–Kontorova problem.
At larger β steady interface propagation becomes possible in certain subcritical
velocity intervals. For instance, at β = −0.75 we found admissible traveling wave
solutions in the intervals: [0.24, 0.5] (between V1 = 0.215 and c = 0.5); [0.142, 0.19]
(between V1 and V2 = 0.1279); [0.1, 0.11] (between V2 and V3 = 0.0912); [0.078, 0.08]
(between V3 and V4 = 0.0708); and possibly in some shorter intervals at smaller V .
Two such solutions are shown in Figure 7.5. The first one corresponds to V = 0.16,
which is between the first and second resonances; unlike its counterpart at β = −0.2,
this solution is admissible. The second admissible profile corresponds to the value
of velocity V = 0.105 located between the second and third resonances. In this case
there are five radiative modes, two in front and three behind the phase boundary. The
appearance of the small-velocity intervals of existence of the traveling wave solutions
548
LEV TRUSKINOVSKY AND ANNA VAINCHTEIN
V =c
G
1.5
1
G(0)
0.2
0.4
0.6
0.8
V
Fig. 7.6. Kinetic relation G(V ) for β = −1/8. The region inside the rectangle should be
excluded because the corresponding solutions violate admissibility constraints (3.5).
at nonzero β is due to the presence of P -roots at nonzero β: as β grows in absolute
value, these roots move closer to the real axis, widening the transition layer and
suppressing oscillations due to the Q-roots (see Figure 7.3).
7.3. Kinetic relation. Using (6.5) and the known dispersion spectrum, we can
now explicitly evaluate the kinetic relation G(V ). A representative example is shown
in Figure 7.6. At resonance velocities the configurational force required to move the
interface tends to infinity. The singularities are due to equal curvatures of the energy
wells ensuring that the energy transport is simultaneously blocked in both phases.
The main physical reasons are related to low dimensionality of the model and the
absence of microscopic dissipation.
As we discussed above, at sufficiently small β the entire region around the smallvelocity resonances has to be excluded because the corresponding solutions violate
the admissibility constraints (3.5). With β increasing, some of the small-velocity
solutions between the resonances become admissible, as shown in Figure 7.7(b)–(d).
Observe also that there is an infinite number of β at which the sonic speed c coincides
with one of the resonance velocity. Thus, at β = −0.9539 we have c = V1 = 0.2147,
implying that for β ≤ −0.9539 the subsonic region lies below the first resonance (see
Figure 7.7(d)). Overall, the total domain of existence of the traveling wave solutions
shrinks as β → −1, while the domain of admissible traveling waves between the
resonances expands.
Zero-velocity limit. To check the compatibility of static and dynamic branches
of solutions it is instructive to trace the zero velocity limit of our dynamic theory.
At V = 0 we can use (5.2) with F ± = {±iλ}, where λ is defined in (7.5). After
some algebraic manipulations, the family of lattice-trapped equilibria (5.2) can be
represented in the form
⎧
σ+1
eλ(n+1/2)
⎪
⎪
, n < 0,
−
⎪
⎨ 1+β
2(1 + β) cosh(λ/2)
wn =
(7.6)
⎪
e−λ(n+1/2)
⎪
⎪ σ +
⎩
, n ≥ 0,
1+β
2(1 + β) cosh(λ/2)
where σ lies in the region (5.4). The solutions (7.6) can be shown to be metastable
[32]. The expression for the Peierls stress (5.5) marking the threshold of metastability
can now be written explicitly as
1
1 + β.
σP =
2
549
KINETICS OF PHASE TRANSITIONS
(a)
b=-0.2
(b)
G
G
2.5
2.5
2
2
1.5
1.5
1
b=-0.5
1
G(0)
G(0)
0.5
x
0.2
0.4
0.6
(c)
0.8
V
0.2
0.4
0.6
0.8
V
0.8
V
(d) b=-0.96
b=-0.75
G
G
G(0)
2.5
2
2
1.5
1.5
G(0)
1
0.5
0.5
0.2
0.4
0.6
0.8
V
0.2
0.4
0.6
Fig. 7.7. Kinetic relations G(V ) for different β. The point marked by x in (a) corresponds to
the numerical results in Figures 8.1 and 8.2.
Observe that at β = 0 the Peierls stress coincides with the spinodal stress σS = 1/2. As
β grows, the trapping region becomes narrower and eventually disappears at β = −1
(σP = 0). The upper boundary of the trapping region (5.4) corresponds to the case
when w0 = wc , which is exactly the condition (3.4). The corresponding saddle-point
configuration is given by
⎧
λ/2
− eλ(n+1/2)
⎪
⎪ wc + e
, n < 0,
⎪
⎨
2(1 + β) cosh(λ/2)
(7.7)
wn = lim w(n − V t) =
V →0
⎪
e−λ(n+1/2) − e−λ/2
⎪
⎪
⎩ wc +
, n ≥ 0.
2(1 + β) cosh(λ/2)
Using this solution, we can obtain the value of the configurational force at the depinning limit:
(7.8)
G(0) =
1
1
(w− + w+ ) − wc = √
.
2
2 1+β
Notice that although the Peierls stress tends to zero when β → −1, the corresponding
value of the configurational force G(0) tends to infinity. This is, of course, an artifact
of our specific assumptions concerning the elastic moduli and is due to the divergence
of the macroscopic transformation strain in the limit.
Sonic limit. The qualitative behavior of the function G(V ) in the limit V → c
depends on β. If −1/4 < β ≤ 0, and V c, the wave spectrum contains a single
wave number k which approaches zero as V → c. Expanding the expression for the
configurational force (6.5) at small k we obtain
G∼
6
,
(1 + 4β)k 2
550
LEV TRUSKINOVSKY AND ANNA VAINCHTEIN
xn
shock
1240
1220
phase boundary
1200
shock
1180
10
20
30
40
t
Fig. 8.1. Positions xn (t) of every fifth particle in the interval 290 ≤ n ≤ 345 in numerical
solution of the Riemann problem with initial data compatible with the traveling wave at V = 0.375
at β = −0.2. The phase boundary is initially placed at n0 = 300, and the problem is solved on
the interval 1 ≤ n ≤ 600. The corresponding point on the kinetic relation G(V ) is marked by x in
Figure 7.7a.
which implies that G(V ) → ∞ as V → c (see Figures 7.6 and 7.7(a)). The picture
is qualitatively different when −1 < β < −1/4. In this case as V approaches c from
below, the limit of the corresponding real wave number k is nonzero ks , and therefore
configurational force G(V ) remains finite (see Figure 7.7(b)–(d)).
8. Stability of the traveling waves. We now present some numerical experiments aimed at accessing stability of the admissible traveling wave solutions. To
simplify the consideration of the transient regimes for the system (2.8), we consider
Riemann initial data of the form
⎧
0
⎪
⎨ (w− , 0), n < n0 ,
(wc , 0), n = n0 ,
(wn , vn )t=0 =
(8.1)
⎪
⎩
0
(w+
, 0), n > n0 .
0
0
We assume that w+
< wc and w−
> wc . The analysis of the corresponding continuum
problem for the p-system (1.2) suggests the formation of a phase boundary with two
shocks in front and behind (e.g., [17]). This is indeed what we see in Figure 8.1, which
shows a typical numerical solution of (2.8) obtained using the Verlet algorithm on a
large domain.
After a transient period, the phase boundary starts moving with a constant speed.
To check convergence of the non-steady-state problem to the admissible traveling wave
solution we used the following algorithm. For a traveling wave solution with velocity
V , (4.9) provides the average strains at both sides of the discontinuity w± (V ). We
can then use (4.13) and the Rankine–Hugoniot jump conditions across the shocks to
0
compute the corresponding initial strains w±
in terms of V and v+ . Without loss of
0
generality, we choose v+ so that w+ = 0 and obtain an explicit formula relating the
Riemann data with the observed phase boundary velocity:
0
w−
(V ) = w− (V ) + w+ (V ) +
V
=
2
wc +
c(c2 − V 2 )
k∈Npos (V )
4 sin2 (k/2)
|kLk (k, V )|
+
c(c2
V
.
− V 2)
551
KINETICS OF PHASE TRANSITIONS
(a)
(b)
wn
2.5
wn
shock
2
2
1.5
phase boundary
wc
100
0.5
phase I
0
1.5
phase II
200
300
400
500
shock
n
320
340
360
380
n
0.5
Fig. 8.2. (a) Strain profile wn (200) for the numerical solution shown in Figure 8.1. (b) The
same solution (solid line) zoomed in around the phase boundary (inside the rectangle in (a)) and
compared to the analytical traveling wave solution (thick dashed line).
Computations based on this relation consistently indicate that the fast branch of the
kinetic relation (V > V1 ) corresponds to traveling wave solutions with a finite domain
of attraction. An example corresponding to β = −0.2 and V = 0.375 is shown in
Figures 8.1 and 8.2. One can see not only that the generated phase boundary propagates steadily with the predicted velocity but that the strain profile wn (t) zoomed
around the phase boundary (solid line in Figure 8.2(b)) compares perfectly with the
analytical solution (4.3) (thick dashed line). The above analysis suggests that the fast
branch of the kinetic relation with V > V1 , corresponding to traveling wave solutions
with a single oscillatory mode behind and monotonic leading edge, is locally stable;
proving this conjecture rigorously is highly nontrivial.
Unfortunately, we were not able to find similar evidence of numerical stability for
admissible traveling waves with V < V1 exhibiting oscillations both behind and in
front of the phase boundary. Numerical simulation for the data expected to converge
to the particular traveling wave lead instead to a solution which does not agree with
the traveling wave ansatz although the phase boundary propagates steadily (with a
slightly smaller velocity). The macroscopic solution is very close to the corresponding
traveling wave; the analytical formula captures the core structure but not the oscillations. One can conjecture that while an admissible traveling wave with radiation on
both sides of the front is not a global attractor, it is surprisingly close to one.
Finally, we refer to [17] for a related study of the phase boundary stability in the
context of a different discretization of the mixed-type p-system.
9. Conclusions. In this paper we used a physically motivated discretization of
the p-system to derive an explicit kinetic relation for a one-dimensional theory of
martensitic phase transitions. The macroscopic dissipation was interpreted as the
energy of the lattice waves emitted by a moving macroscopic discontinuity. By using
the simplest piecewise linear model, we obtained an explicit formula for the continuum rate of entropy production which depends only on interatomic potentials. We
showed that despite the difference in the structure of micro and macro theories, the
assessments of dissipation at different scales are fully compatible. The present study
complements previous analyses of related systems in fracture and plasticity framework
by including general harmonic long-range interactions.
More specifically, we showed that contrary to the simplest theory with NN interactions, which has a mean field character and is therefore degenerate, strongly nonlocal
models produce multivalued kinetic relations with several admissible branches and
552
LEV TRUSKINOVSKY AND ANNA VAINCHTEIN
rich variety of configurations of emitted lattice waves. In addition to enlarging the
domain of existence of steady-state regimes, sufficiently strong long-range interactions
significantly alter the structure of mobility curves near sonic speeds. The nonlocality
also affects the size of lattice trapping: as long-range interactions become stronger,
the trapping region reduces in size in terms of stress. At the same time, it widens
in our model in terms of driving forces, which emphasizes an important difference
between the physical and configurational descriptions. Although the main effects of
nonlocality were illustrated in the paper by the explicit computations for the NNN
model, we have also conducted a similar study of the NNNN model which showed
qualitatively similar behavior.
The present work was motivated by similar studies of the semilinear discrete
Frenkel–Kontorova model (e.g., [11, 12]). While the two models turn out to be equivalent in static and overdamped limits [32, 34], the fully inertial versions are quite
different. An additional level of complexity in the quasilinear model considered here
is associated with a different structure of nonlinearity that results in the presence of
the limiting characteristic velocity, microscopic and macroscopic particle velocities,
and the discrete Rankine–Hugoniot jump conditions.
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